ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS Yun, Jae Heon;
In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.
relaxation iterative method;semi-convergence;pseudo-spectral radius;singular saddle point problem;scaled preconditioner;
M. Arioli, I. S. Duff, and P. P. M. de Rijk, On the augmented system approach to sparse least squares problems, Numer. Math. 55 (1989), no. 6, 667-684.
Z.-Z. Bai, G. H. Golub, and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004), no. 1, 1-32.
Z.-Z. Bai, B. N. Parlett, and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005), no. 1, 1-38.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
Z. Chao and G. Chen, Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems, Appl. Math. Lett. 35 (2014), 52-57.
Z. Chao, N.-M. Zhang, and Y.-Z. Lu, Optimal parameters of the generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 266 (2014), 52-60.
H. C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput. 20 (1999), no. 4, 1299-1316.
H. C. Elman and D. J. Silvester, Fast nonsymmetric iteration and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput. 17 (1996), no. 1, 33-46.
B. Fischer, A. Ramage, D. J. Silvester, and A. J. Wathen, Minimum residual methods for augmented systems, BIT 38 (1998), 527-543.
G. H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), no. 3, 71-85.
F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incom-pressible flow of fluid with free surface, Phys. Fluids 8 (1965), 2182-2189.
J.-I. Li and T.-Z. Huang, The semi-convergence of generalized SSOR method for singular augmented systems, High Performance Computing and Applications, Lecture Notes in Computer Science 5938 (2010), 230-235.
G. H. Santos, B. P. B. Silva, and J.-Y. Yuan, Block SOR methods for rank deficient least squares problems, J. Comput. Appl. Math. 100 (1998), no. 1, 1-9.
S. Wright, Stability of augmented system factorization in interior point methods, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 191-222.
S.-L. Wu, T.-Z. Huang, and X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (2009), no. 1, 424-433.
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
J.-Y. Yuan and A. N. Iusem, Preconditioned conjugate gradient methods for generalized least squares problem, J. Comput. Appl. Math. 71 (1996), no. 2, 287-297.
J. H. Yun, Variants of the Uzawa method for saddle point problem, Comput. Math. Appl. 65 (2013), no. 7, 1037-1046.
J. H. Yun, Convergence of relaxation iterative methods for saddle point problem, Appl. Math. Comput. 251 (2015), 65-80.
G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 219 (2008), no. 1, 51-58.
G.-F. Zhang and S.-S. Wang, A generalization of parameterized inexact Uzawa method for singular saddle point problems, Appl. Math. Comput. 219 (2013), no. 9, 4225-4231.
N. Zhang, T.-T. Lu, and Y. Wei, Semi-convergence analysis of Uzawa methods for singular saddle point problems, J. Comput. Appl. Math. 255 (2014), 334-345.
N. Zhang and Y. Wei, On the convergence of general stationary iterative methods for range-Hermitian singular linear systems, Numer. Linear Algebra Appl. 17 (2010), no. 1, 139-154.
B. Zheng, Z.-Z. Bai, and X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 431 (2009), no. 5-7, 808-817.